A level Maths mechanics
Mechanics makes up a section of A level Maths. It rewards clear diagrams and careful use of a few core ideas.
On this page
Kinematics
- Constant acceleration: the SUVAT equations.
- Variable acceleration: use calculus. v = ds/dt, a = dv/dt, and integrate to go back.
- Velocity-time graphs: gradient is acceleration and area is displacement.
Forces and Newton's laws
- F = ma, applied to the resultant force in each direction.
- Weight W = mg, normal reaction, tension and thrust.
- Resolve forces into perpendicular components on slopes and for forces at angles.
Friction
Friction is at most μR, where R is the normal reaction. When an object is moving or on the point of moving, friction equals μR and acts against the motion.
Connected particles
For pulleys and towing problems, write F = ma for each particle separately. The tension is the same throughout a light, inextensible string over a smooth pulley.
Moments
A moment is force × perpendicular distance from the pivot. For a body in equilibrium, the resultant force is zero and moments about any point balance.
Projectiles
Split motion into horizontal (constant velocity) and vertical (acceleration g downward). Time links the two.
Exam tips
- Always draw a clear force diagram.
- State the direction you take as positive.
- Give answers to a sensible accuracy when g = 9.8 is used.
- State modelling assumptions, such as a particle, a light string or a smooth surface, when asked.
Thinking like a modeller
Mechanics is about modelling the real world with simple assumptions. A box becomes a particle, a rope becomes a light inextensible string, and a surface becomes smooth or rough. Exam questions often ask you to state or criticise these assumptions, so learn what each one means: a particle has no size, so it cannot rotate; a light string has no mass, so the tension is the same throughout; a smooth surface has no friction.
Worked examples
Example 1: variable acceleration
A particle has velocity v = 3t² − 2t m/s and starts at the origin. Find its displacement at t = 2.
- s = ∫(3t² − 2t) dt = t³ − t² + c, and s = 0 at t = 0 gives c = 0.
- At t = 2: s = 8 − 4 = 4 m.
Example 2: a lift
A 70 kg person stands in a lift accelerating upwards at 1.2 m/s². Find the reaction from the floor. Use g = 9.8 m/s².
- R − 70g = 70 × 1.2, so R = 70(9.8 + 1.2) = 770 N.
Example 3: connected particles
A 3 kg block on a smooth table is joined over a smooth pulley to a hanging 2 kg block. Find the acceleration and the tension.
- For B: 2g − T = 2a. For A: T = 3a.
- Adding: 2g = 5a, so a = 3.92 m/s² and T = 3 × 3.92 ≈ 11.8 N.
Example 4: a rough slope
A 5 kg block slides down a slope at 30° with coefficient of friction 0.3. Find its acceleration.
- R = 5g cos30°, friction = 0.3R.
- a = g(sin30° − 0.3 cos30°) = 9.8(0.5 − 0.260) ≈ 2.35 m/s².
Example 5: moments
A uniform 4 m beam of weight 60 N rests on supports at each end. A 40 N load sits 1 m from the left end. Find the reactions.
- Moments about the left support: Rright × 4 = 60 × 2 + 40 × 1 = 160, so Rright = 40 N.
- Vertical forces: Rleft = 100 − 40 = 60 N.
Common mistakes
- Missing a force on the diagram, especially friction or the normal reaction.
- Using SUVAT when acceleration varies with time.
- Taking moments with distances that are not perpendicular to the force.
- Giving answers to more significant figures than g = 9.8 justifies.
Practice questions
Try each one before opening the answer.
Q1. A car of mass 1200 kg accelerates at 2 m/s² against a resistance of 600 N. Find the driving force.
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Q2. A particle's displacement is s = 2t³ − 9t² + 12t. When is it instantaneously at rest?
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Q3. A 2 kg box rests on a rough horizontal floor (μ = 0.4). Find the largest horizontal force that will not move it.
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Frequently asked questions
Which value of g should I use?
Use the value in the question. If none is given, A level Maths usually uses 9.8 m/s².
Is projectile motion in A level Maths?
Yes, in the second year. Treat horizontal and vertical motion separately and link them with time.
Key takeaways
- Draw a clear force diagram for every problem.
- Use SUVAT only for constant acceleration; otherwise use calculus.
- Apply F = ma to each particle in connected systems.
- For equilibrium, resolve forces and take moments.
- State and understand modelling assumptions.
Where to go next
These guides support the mechanics section.
A method that works for most mechanics questions
- Draw a large, clear diagram and mark every force, with arrows in the right directions.
- Choose a positive direction and stick to it.
- Resolve forces parallel and perpendicular to the motion or the slope.
- Write F = ma, or "forces balance" for equilibrium, in each direction.
- Solve, then check the answer makes physical sense, such as a positive tension.
Following the same steps every time keeps you calm in the exam, and the diagram alone often earns a mark.
Projectiles and friction in Year 2
Second-year mechanics adds projectiles, friction on slopes, and moments with non-parallel forces. For projectiles, resolve the initial velocity into horizontal and vertical parts, use SUVAT vertically with acceleration g downwards, and use distance equals speed times time horizontally. Time is the link between the two directions. For friction, remember that friction is at most μR; it only equals μR when the object is moving or on the point of moving, which the question will usually describe as limiting equilibrium.
Ladder problems combine both ideas: take moments about the foot of the ladder, resolve horizontally and vertically, and use the friction limit to find when it is about to slip.
With a reliable method and plenty of practice, mechanics often becomes the most predictable and rewarding part of the whole paper, and a reliable source of marks.